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Results tagged with lie-algebras
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user 5301
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
8
votes
What is the relation between characters of a group and its lie algebra?
A formal character of a finite-dimensional representation can be, indeed, thought of as a trace function. You can define it on Cartan subalgebra, maximal torus, the set of semsimple elements of the co …
- 12.1k
2
votes
Purely algebraic proof for unitarizability of representations of a compact real semisimple L...
Why not, doc? Take a unitary representation $V$ of $G$. Its tensor power $T^nV$ is unitary as well via the obvious form
$$
<a\otimes b\otimes \ldots , a^\prime \otimes b^\prime \ldots> =
<a,a^\prime> …
- 12.1k
4
votes
0
answers
221
views
Second symmetric square of the adjoint representation
I have just come across the following experimental fact. Let ${\mathfrak g}$ be a simple complex Lie algebra.
Fact: ${\mathfrak g}$ is a constituent of $S^2{\mathfrak g}$ if and only if ${\mathfrak …
- 12.1k
3
votes
Accepted
Irreducibility of the $\mathfrak{g}$-module $\mathfrak{o}(k)/ad(\mathfrak{g})$
Note that ${\mathfrak o}(k)\cong \wedge^2 {\mathfrak g}$. It has ${\mathfrak g}$ as a summand, coming from the Lie bracket $\wedge^2 {\mathfrak g} \rightarrow {\mathfrak g}$ . Calculation of the rest …
- 12.1k
5
votes
Accepted
adjoint action of a Levi subalgebra
Not sure that there would be a closed formula as Jim has pointed out. You can easily calculate it in examples because you can determine highest weight vectors in $g$ for $m$. Indeed, if $\alpha_1, \ld …
- 12.1k
1
vote
When is a nilpotent Lie algebra isomorphic to the associated graded of its lower central ser...
$\DeclareMathOperator\gr{gr}$Too tired to think clearly, but it looks like a standard Deformation Theory thingy.
We have natural linear maps $\gamma_n (L)\rightarrow \gr_n L$. Split them as linear map …
- 12.1k
1
vote
Low dimensional nilpotent Lie algebras
In his 1957 paper Dixmier computes the centres of the universal enveloping of all f-d complex Lie algebras up to dimension 5 and, in particular, lists them all.
- 12.1k
0
votes
Accepted
Conditions on $\beta$ under which the trace pairing restricted to $\mathfrak{so}(V,\beta)$ i...
Yes to all three questions. By diagonalisation of forms, WLOG, $\beta$ is diagonal with $\pm 1$ on the main diagonal. Now let us just compute (and you can get an explicit formula for your transpositio …
- 12.1k
1
vote
Accepted
Representations of reductive Lie group
You need to be over a field of zero characteristic and your representation needs to be rational, i.e. matrix entries need to be algebraic functions on $G$. Then it is completely reducible, see any boo …
- 12.1k
3
votes
Introduction to W-Algebras/Why W-algebras?
The motivation did not change. The theory of finite W-algebras advanced quite a bit, imho, mostly thanks to the works of Losev.
There is a good introductory text by Arakawa now.
- 12.1k
4
votes
Representations are determined by characters : Groups and Lie algebras
Doc, I am not sure what your question is, but the answer is yes. Whatever definition of character you are using, any two extensions of $M$ by $N$ will have the same character. Thus, a non-trivial exte …
- 12.1k
2
votes
Building Lie-like algebras from modules over semisimple Lie algebras
The general idea smells like coloured Lie superalgebras. In a nutshell, take the category of $\Gamma$-graded vector spaces and skew braiding by a bicharacter of $\Gamma$. Now consider Lie algebras in …
- 12.1k
3
votes
Which is the correct universal enveloping algebra in positive characteristic?
You forgot the third one: restricted enveloping algebra. Hence, in characteristic p we have 3 enveloping algebras with homomorphisms
U->U_0->U_{dp}
All 3 are Hopf algebras and can be used for differen …
- 12.1k
5
votes
Accepted
Killing form vs its counterpart in a given represenation
They are proportional if $g$ is simple. The form $K_\phi$ defines a homomorphism from the adjoint to the coadjoint representation. If the adjoint representation is irreducible, i.e. $g$ is simple, you …
- 12.1k
1
vote
Complete reducibility and field extension
Amazingly, I cannot see an elementary solution. I believe there should be one.
Otherwise, one can expand the comment of YCor with some standard Ring Theory. Let $A$ be the image of $U(L)$ in $End_F(V …
- 12.1k
3
votes
Accepted
The Lie algebra of the subgroup of $GL(n)$ preserving a given variety
Doc, you are right but only amorally. You need to replace the tangent vectors with jets to capture the behavior of your cone.
Let $I(Y)$ be the ideal of zeroes of your $Y$. Then
$$
Lie (G_Y) = \{ X …
- 12.1k
13
votes
4
answers
5k
views
What is the universal enveloping algebra?
Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). W …
- 12.1k
4
votes
Is there a canonical Hopf structure on the center of a universal enveloping algebra?
I would be very surprised if the answer is YES, in general. This would turn the spectrum of Z(g) into an algebraic group, and, in particular, force it to be smooth (if Z(g) is finitely generated). Qui …
- 12.1k
5
votes
Lie Algebras and Simple Connectivity for general algebraic groups
No, it does not. The additive $G$ and the multiplicative $H$ groups have isomorphic Lie algebras but only trivial group homomorphisms between them.
The Lie algebra of $G$ has a restricted structure! …
- 12.1k
4
votes
Non-cosemisimple duals of pointed Hopf algebras
No way, doc! Take a finite $p$-group $G$. Let ${\mathbb F}$ be a field of characteristic $p$. The group algebra ${\mathbb F}G$ is as pointed as it gets. But its dual ${\mathbb F}G^{\ast}$ is not cosem …
- 12.1k
5
votes
Accepted
Hopf structure on the universal enveloping of a super Lie algebra
No, it is not true unless the odd part is zero or the characteristic is 2. More precisely, there is no natural Hopf algebra structure outside these conditions, but there may be some odd examples. It i …
- 12.1k
13
votes
Geometric interpretation of Universal enveloping algebras
You can think of $Ug$ as the algebra of distributions on $G$ supported at 1. Alternatively, you can think of $Ug$ as differential operators, for instance as global sections of monodromic differential …
- 12.1k